Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexlimddvcbv Structured version   Visualization version   GIF version

Theorem rexlimddvcbv 44197
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44195. The equivalent of this theorem without the bound variable change is rexlimddv 3159. Usage of this theorem is discouraged because it depends on ax-13 2375, see rexlimddvcbvw 44196 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
rexlimddvcbv.1 (𝜑 → ∃𝑥𝐴 𝜃)
rexlimddvcbv.2 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
rexlimddvcbv.3 (𝑥 = 𝑦 → (𝜃𝜒))
Assertion
Ref Expression
rexlimddvcbv (𝜑𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦   𝜒,𝑥   𝜃,𝑦   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)

Proof of Theorem rexlimddvcbv
StepHypRef Expression
1 rexlimddvcbv.1 . 2 (𝜑 → ∃𝑥𝐴 𝜃)
2 rexlimddvcbv.3 . . 3 (𝑥 = 𝑦 → (𝜃𝜒))
3 rexlimddvcbv.2 . . 3 ((𝜑 ∧ (𝑦𝐴𝜒)) → 𝜓)
42, 3rexlimdvaacbv 44195 . 2 (𝜑 → (∃𝑥𝐴 𝜃𝜓))
51, 4mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator